Homogeneous Riemannian manifolds with non-trivial nullity

ANTONIO J. DISCALA, CARLOS OLMOS, FRANCISCO VITTONE

TRANSFORMATION GROUPS

BIRKHAUSER BOSTON INC

Lugar: Boston; Año: 2022 vol. 27 p. 31 - 72

1083-4362

We develop a general theory for irreducible homogeneous spaces M=G/H, in relation to the nullity ν of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e. where the nullity distribution is not parallel. Moreover, we construct irreducible examples of conullity k=3, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that H is trivial and G is solvable if k=3.Another of our main results is that the leaves of the nullity are closed (we used a rather delicate argument). This implies that M is a Euclidean affine bundle over the quotient by the leaves of ν. Moreover, we prove that ν⊥ defines a metric connection on this bundle with transitive holonomy or, equivalently, ν⊥ is completely non-integrable (this is not in general true for an arbitrary autoparallel and flat invariant distribution).We also found some general obstruction for the existence of non-trivial nullity: e.g., if G is reductive (in particular, if M is compact), or if G is two-step nilpotent.